Keir Starmer cries as he quits No 10 claiming a deluded list of'achievements' - now Britain awaits its seventh PM in ten years Putin'prepares mass call-up' for the Ukraine meat-grinder - as video shows Russian veteran with no legs threatening recruiter with a knife in sign of growing resistance facing desperate Kremlin'Al Roker is an absolute ****': KENNEDY's Today show insider gives brutal behind-the-scenes verdict on beloved weatherman and names other two-faced NBC hosts No one can see the real reason Jelly Roll divorced Bunnie XO. Boston's Scotland-loving residents claim England fans are'ruining the vibe' compared to the Tartan Army Secret life of John Travolta's daughter Ella Bleu: New details about'unusual' relationship with her dad revealed by insiders amid fears that aspiring actress is'stuck' Johnny Depp's ex Amber Heard gives rare glimpse of daughter Oonagh, five, after finishing 10k race in Spain Colorado siblings VANISH from home in middle of the night... and police ...

High-stakes Iran war talks get underway as JD Vance is joined by pregnant wife Usha in leading familiar team of American negotiators... but fresh conflict threatens to derail peace plan Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt'Media-obsessed' Anna Paulina Luna reveals secret to her rising power as she turns into Republicans' 'favorite headache' Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN No one can see the real reason Jelly Roll divorced Bunnie XO. Royals wish Prince William happy birthday and Father's Day with sweet photo of him and Charlotte after King's Trooping the Colour - as Charles pays tribute to Philip Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Lauren Sanchez swaps sultry dresses for leggings as she flaunts her all-natural beauty in rare glam-free photo in honor of Father's Day'I married a death row double killer': London single mother tells the full, incredible story of how she wed a Texas convict, then watched his execution Americans are flocking to'Goldilocks' city that has all the perks of major southern hubs... but homes are a fraction of the price The four mistakes that led to bungee tragedy on Skeleton Bridge: FRED KELLY saw the scene for himself, now he retraces the prelude to disaster. So was it really an accident? Ashen-faced minister admits Starmer is facing'political reality' as PM is expected to quit tomorrow Trump says algae-infested Reflecting Pool must be EMPTIED for repairs as knife-wielding'vandals' tear hole in facade and destroy $16 million renovation Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Horrific'womb-raider' murder that shocked US: Boyfriend and family's astonishing account of how woman butchered her friend to steal her unborn child after lying SHE was pregnant I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips.

Trump says algae-infested Reflecting Pool must be EMPTIED for repairs as knife-wielding'vandals' tear hole in facade and destroy $16 million renovation Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN New York Knicks fan caught in'disgusting' act during team's NBA championship parade celebrations Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. No one can see the real reason Jelly Roll divorced Bunnie XO. Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Wyndham Clark's stunning girlfriend pays tribute to polarizing golfer as he stands on the brink of US Open glory Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Dad jokes are GOOD for you, studies show - so, do these classic gags make you cringe? READ MORE: Why English football chants are catchier than rest of world's They're often seen as cringeworthy, but try not to roll your eyes at your dad's jokes too much this Father's Day.

Trump says algae-infested Reflecting Pool must be EMPTIED for repairs as knife-wielding'vandals' tear hole in facade and destroy $16 million renovation Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN New York Knicks fan caught in'disgusting' act during team's NBA championship parade celebrations Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. No one can see the real reason Jelly Roll divorced Bunnie XO. Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Wyndham Clark's stunning girlfriend pays tribute to polarizing golfer as he stands on the brink of US Open glory Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' With hot weather and glorious sunshine forecast this weekend, many Brits will be excited to fire up the barbecue. But before you light the coals, experts warn that there are some surprising foods you should never cook on there.

Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Former Olympian is arrested for allegedly vandalizing Reflecting Pool... but he claims he merely touched it Embattled Alexi Lalas makes controversial World Cup declaration amid tension with Fox colleagues: 'Makes you look like a weak poser' Cocaine scandal ripping the Hamptons apart: New York elite's dirty secret leaves mothers too afraid to let their children out... as police issue urgent warning Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN Germany vs Ivory Coast - World Cup Group E RECAP: Deniz Undav's second goal seals his nation qualification to the knockouts as he nets winner in second-half stoppage time I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. No one can see the real reason Jelly Roll divorced Bunnie XO. Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Three more arrested over bungee jumper's death after she was hurled from bridge without a rope Ex-partner of dad who was berated for taking his daughters into women's bathroom claims he'exploited' girls and accuses him of failing to pay child support... before he hits back Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Candace Owens hits out at nasty rumors claiming she was DEAD... as fellow MAGA influencer claims her account was hacked The four mistakes that led to bungee tragedy on Skeleton Bridge: FRED KELLY saw the scene for himself, now he retraces the prelude to disaster. So was it really an accident?

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the topological descriptors - the Euler characteristic (EC) and persistent homology (PH). In particular, we establish a complete characterization of the expressive power of EC on general color-based filtrations. We also show that the PH descriptors of (virtual) graph products contain strictly more information than the computation on individual graphs, whereas EC does not. Additionally, we provide algorithms to compute the PH diagrams of the product of vertex-and edge-level filtrations on the graph product. We also substantiate our theoretical analysis with empirical investigations on runtime analysis, expressivity, and graph classification performance. Overall, this work paves way for powerful graph persistent descriptors via product filtrations.

Terrifying stomach cancer explosion sweeps the US: After fitness influencer's shock death, experts reveal subtle early signs that are too often ignored... and lifestyle tweaks that can PREVENT it Actress, 43, announces she is expecting with sweet video after detailing'complicated' journey to motherhood and hope of having third child Trump foe Rosie O'Donnell to replace Jimmy Kimmel as he steps back from his show Deadly secrets of gorgeous California enclave where college girls were killed by a'sneaker'... now experts say they could have been SAVED The other women left devastated by Jelly Roll's divorce: Why his daughter is now'disgusted'... as Bunnie's baby bombshell rocks Nashville The shaming of America's original mommy influencer after tragedy that divided the nation: Bode Miller's wife Morgan breaks cover to reveal agonizing regret that still haunts her since daughter's drowning Trump boasts there's'no limits' to his power and posts bizarre memo by fake historian comparing him to Hitler More young Americans are living with their parents than ever before... and there is a shocking reason behind the boomerang trend I was mortified when my husband always said no to sex. Then I realised the mistake I was making. This is the change that's completely transformed marital love-making in middle age: ALICE SNAPE Revealed: Hero, 24, who saved man's LIFE in dramatic rescue during New York Knicks victory parade after defying cops' orders: 'I'm just another New Yorker' REVEALED: Gavin Newsom steered millions of dollars of donations to nonprofits connected to his wife... as Trump's DOJ probes couple The shingles vaccine could lower dementia risk'by up to a quarter' - but scientists are still puzzled why Farce of Obama's $850m'monstrosity': As clucking liberal elite cheer Barack's grand opening, outraged Chicago locals tell HARRIET ALEXANDER awkward truth about library Why turnips MUST be in your grocery cart if you're trying to lose weight Taco Bell's finally fixes a glaring menu gap - and brings back a fan favorite after years Mom thought popular'natural' health supplement was safer than Xanax. She took it... then never woke up. Don't make the same mistake Mother and child in critical condition after being swallowed into ocean by ANOTHER monstrous California wave... just days after college students were killed by breaker'I was taken from school and trained to fly UFOs with my mind,' claims child genius A former gifted child has come forward with claims that he was removed from public school and secretly trained to develop psychic abilities for military and UFO-related applications.

Gilgo Beach serial killer Rex Heuermann's ex-wife reacts to his sentencing as monster who killed eight women is transferred to new prison to begin life behind bars Boy, three, 'attacked by at least one crocodile' after being'thrown into zoo pit by man with learning difficulties who broke away from carers' - as suspect'not fit for interview' is bailed Jelly Roll stops concert to respond to wife Bunnie XO's bombshell podcast on their divorce Hegseth puts NATO on notice as he launches review of US troops in Europe and blasts allies for'shameful' behavior I was mortified when my husband always said no to sex. Then I realised the mistake I was making. This is the change that's completely transformed marital love-making in middle age: ALICE SNAPE Mom thought popular'natural' health supplement was safer than Xanax. She took it... then never woke up. Don't make the same mistake JD Vance turns on Israeli allies who are criticizing Trump's Iran deal: 'Wake up and smell reality' The other women left devastated by Jelly Roll's divorce... why his daughter is now'disgusted'... and Bunnie XO's one red-line demand before she would agree to the split Joe Biden mumbles to himself and requires stage direction as he aimlessly wanders off at Obama's library debut Tourists run for their lives as gunfire erupts in New York's Times Square as terrified parents drag children to safety Heartbroken family of college girls who drowned dispute account of their final moments before they were swept out to sea as they mourn'responsible and kind' students Oscar-winning director's daughter and her husband's deaths'medically related' as cops give grim update after couple were found in SUV on California highway Furious woke woman storms out of restaurant because customers were singing National Anthem ...and vows never to return A bold new experiment to streamline how Americans buy new cars... and auto dealerships are already scared Secret White House blacklist leaked by insider: 'Worst' influencers named and shamed... as foul-mouthed backstabbing erupts Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Bill Clinton's VERY cozy moment with Michelle while Hillary looks the other way... and the best UNSEEN moments from Obama public library opening Farce of Obama's $850m'monstrosity': As clucking liberal elite cheer Barack's grand opening, outraged Chicago locals tell HARRIET ALEXANDER awkward truth about library Humiliating new joke about Trump that's the talk of Washington... as White House moles tell me there's more to this story than meets the eye: MARK HALPERIN Humanoid robot is spotted BEGGING on a street in China - claiming it has'no money to recharge' READ MORE: China unveils the world's first self-driving TOILET While many people worry that robots are coming to take their jobs, one unlucky bot seems to have fallen on hard times.

Embattled Gavin Newsom's stunning confession to Justin Trudeau caught on camera at World Cup when he thought no one was watching Tragic final moments of Hollywood legend's daughter and her husband revealed before being mysteriously found dead in their running SUV Boy, three, is thrown into crocodile enclosure at zoo: Man, 30, 'not known to him' arrested on suspicion of attempted murder Haunting final video of beloved Bay Area coffee shop owner, 52, who vanished without a trace: Investigator reveals'unnerving' new clues found inside her home Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Malia and Sasha Obama exude confidence as they attend grand opening of dad Barack's library Mystery surrounds JD Vance's dash to Switzerland as world holds breath for Iranians to confirm peace deal All my friends are suddenly getting divorced. Mid-life wives share taboo sex confessions about why they really leave... including common position that made one hate her husband: JANA HOCKING Taylor Swift's bottomless thirst for attention, her greed and sheer tackiness are now truly unbearable... this latest stunt has shown her true colors: MAUREEN CALLAHAN Male Israeli hostage sexually assaulted by Hamas captor describes multiple attacks he suffered - blindfolded and stripped naked at knifepoint... and'brutal' 20-minute ordeal World Cup's'sexiest fan' sends fans wild with barely-there outfit to watch her country's first game in Dallas Troubling new psychological side effect of Ozempic revealed... as users battle'ghost fat' after losing weight Infection found in wildlife evolved to spread between humans, experts fear... after two clusters are identified Sensational REAL reason Jelly Roll is divorcing Bunnie XO: Insiders reveal'preacher's wife' bombshell that's the talk of Nashville... truth about legendary rocker cuckolding rumor... and G-string mishap Fears of the'Big One' - an earthquake so big it devastates all of California - have risen to new heights after a disturbing discovery under America's most dangerous fault line. Researchers from the US and Switzerland revealed that the San Andreas Fault has reached its highest levels of stress in 1,000 years - adding that it has been more than 160 years since the giant crack in the Earth's crust had a major release of energy . The San Andreas is an 800-mile-long fault line which runs under most of California, passing by Los Angeles in the south and San Francisco in the north and connecting to several other major faults, most notably the San Jacinto Fault near Los Angeles. This connection is where researcher Liliane Burkhard from the University of Hawaiʻi at Mānoa said seismic stress has become so great at the southern end of the San Andreas that a rupture could travel along both fault lines - resulting in a mega quake .

'Ringleader' of alleged UFC drone attack to kill Trump is unmasked as illegal migrant who was granted DACA stay under Obama Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Horrific new videos blow Texas woman's mystery death wide open: Her agonizing'final gasp'... unthinkably vile corpse claims... and sick past of man who saw her last Taylor Swift's bottomless thirst for attention, her greed and sheer tackiness are now truly unbearable... this latest stunt has shown her true colors: MAUREEN CALLAHAN Spy world panic as Tulsi Gabbard prepares to unleash bombshell file dumps on secret CIA'mind control' project and Dr. Fauci Olivia Wilde, 42, complains about being on Maxim's Hot 100 List calling it the'most f***** up thing in the world' Has Taylor Swift already revealed her wedding dress designer? All my friends are suddenly getting divorced. Mid-life wives share taboo sex confessions about why they really leave... including common position that made one hate her husband: JANA HOCKING Kanye West's wife Bianca Censori raises eyebrows in plunging white lace lingerie as she photographs a nude model at Art Basel in Switzerland Knicks set to come face to face with Trump after president was'thunderously booed' at NBA Finals game Sensational REAL reason Jelly Roll is divorcing Bunnie XO: Insiders reveal'preacher's wife' bombshell that's the talk of Nashville... truth about legendary rocker cuckolding rumor... and G-string mishap Teen tourist thrown to death by Central Park horse was trying to save mom who flew out of carriage during family's first visit to Big Apple Father keeps his cool as shouting man calls cops on him for taking his two young daughters into women's restroom Trump privately frets Bibi Netanyahu's zeal to'bomb everyone' could turn him into another disgraced president'Moscow will burn', Zelensky vows as Russia's capital is blanketed in toxic smoke following huge Ukraine drone attack He drove a Rolls-Royce and lived the American dream. But behind the Gucci was the ATF's most unlikely secret weapon. MORE: Meteorologist reveals America's most dangerous cities in super El Niño's'corridor of chaos'... and warns this is only the beginning As many as 40 million people across eight states are in the deadly path of Tropical Storm Arthur after the first named storm of hurricane season made landfall Wednesday night.